A Markov Approach to the Average Run Length of Cusum Charts for an AR(1) Process

  • Ken Nishina
  • Shin’ichiro Matsubara
Conference paper
Part of the Frontiers in Statistical Quality Control book series (FSQC, volume 7)


Some papers have presented the effects of autocorrelated observations to Cusum charts. In this paper, the performance of Cusum charts in a first-order autoregressive (AR(1)) model is examined in some more detail by representing the Cusum statistic by a two-dimensional Markov process. The distribution of Cusum statistic at any time and the Average Run Length (ARL) can be analytically derived. Thus a conditional out of control ARL for any change point can be obtained. The variation pattern of the out of control ARL can be presented until the ARL converges to the steady state ARL. It can be shown that the variation pattern in the case of the no autocorrelation is monotonously decreasing in the change point. In the presence of autocorrelation, the monotonious decrese still holds, however, only when the autocorrelation is not highly positive.


Change Point Control Chart Cusum Chart Positive Autocorrelation Negative Autocorrelation 
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  1. 1.
    Bagshaw, M. and Johnson, R. A. (1975): The Effect of Serial Correlation on the Performance of CUSUM Tests II, Technometrics, Vol. 17, No.l, 73–80.CrossRefMathSciNetGoogle Scholar
  2. 2.
    Brook, D. and Evans, D.A. (1972): “An Approach to the Probability Distribution of Cusum Run Length,” Biometrika, Vol. 59, No. 3, 539–549.CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Goel, A.L. and Wu, S.M. (1971): Determination of ARL and a Contour Nomogram for Cusum Charts to Control Normal Mean, Technometrics, Vol. 13, No. 2, 221–230.MATHGoogle Scholar
  4. 4.
    Goldsmith, P.L. and Whitfield, H. (1961): Average Run Length in Cumulative Chart Quality Control Schemes, Technometrics, Vol. 3, No. 1, 11–20.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Harris, T. J. and Ross, W.H. (1991): Statistical Process Control Procedures for Correlated Observations, The Canadian Journal of Chemical Engineering, Vol. 69, 48–57.CrossRefGoogle Scholar
  6. 6.
    Johnson, R.A. and Bagshaw, M. (1974): The Effect of Serial Correlation on the Performance of CUSUM Tests, Technometrics, Vol. 16, No.l, 103–112.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Lu, C. W. and Reynolds, M. R. (2001): Cusum Charts for Monitoring an Autocorrelated Process. Journal of Quality Technology, Vol. 33, No. 3, 316–334.Google Scholar
  8. 8.
    Nishina, K. (1982): Design of CUSUM Charts in the Presence of Data Correlation, Hinshitu, Vol.12, No.3, 44–53 (in Japanese).Google Scholar
  9. 9.
    Nishina, K. (1986): “Estimation of the Change-point from Cumulative Sum Tests,” Reports of Statistical Application Research, JUSE., Vol. 33, No.4, 1–14.MATHMathSciNetGoogle Scholar
  10. 10.
    Nishina, K. and Wang, P. H. (1996): “Performance of Cusum Charts from the Viewpoint of Change Point Estimation in the Presence of Autocorrelation,” Quality and Reliability Engineering International, Vol. 12, 3–8.CrossRefGoogle Scholar
  11. 11.
    Rowlands, R.J. (1976): Formulae for Performance Characteristica of CuSum Schemes When the Observations are autocorrelated, Ph. D. thesis, University of Wales, Cardiff.Google Scholar
  12. 12.
    Rowlands, R.J. and Wetherill, G.B. (1991): Quality Control, In: Ghosh, B.K. and Sen, P.K. (Eds.) Handbook of Sequential Analysis, New York etc.: Marcel Dekker.Google Scholar
  13. 13.
    Timmer, D. H., Pignatiello, J. and Longnecker, M. (1998): The Development and Evaluation of CUSUM-based Control Charts for an AR(1) process, IIE Transactions, Vol. 30, 525–534.Google Scholar
  14. 14.
    VanBrackle, L.N. and Reynolds, M. R. (1997): EWMA and Cusum Control Charts in the Presence of Correlation, Communication in Statistics — Simulation and Computation, Vol. 26, No. 3, 979–1008.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Ken Nishina
    • 1
  • Shin’ichiro Matsubara
    • 1
  1. 1.Nagoya Institute of TechnologyShowa-ku, NagoyaJapan

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